Kalman Filtering

Kalman filter (KF)-based multi-object tracking (MOT) remains a strong baseline for autonomous driving due to its strong performance, computational efficiency and interpretability. In most practical systems, the process noise and measurement noise covariances are defined globally and shared across object classes, presuming identical uncertainty characteristics across heterogeneous traffic participants. This work revisits this assumption and proposes CANMOT, a class-aware and object-aligned noise modeling framework for KF-based 3D MOT. Class-specific diagonal process and measurement covariance matrices are introduced and optionally expressed in the object coordinate frame to preserve longitudinal-lateral anisotropy. Systematic experiments on the nuScenes benchmark show that class-aware and object-aligned noise modeling improves tracking performance and substantially reduces identity switches compared to state-of-the-art (SotA). In addition, the consistency of the estimated uncertainty is analyzed using the Average Normalized Estimation Error Squared (ANEES) and $χ^2$-based violation tests. The results reveal severe overconfidence in standard KF-based MOT baselines. While the proposed formulation improves calibration without modifying the underlying filtering framework, it still exhibits substantial inconsistency, highlighting the need for further research in this area. Code is available at https://github.com/rst-tu-dortmund/learned-3d-nms.

Kalman filtering performance is highly sensitive to model mismatch and noise covariance tuning. Learning-based approaches address these limitations but typically rely on supervised training with large datasets and do not produce consistent uncertainty estimates. In this paper, we propose a self-supervised Hybrid Adaptive Kalman Filter that learns structured corrections to system dynamics and process noise covariance from measurements alone while preserving the probabilistic structure of the filter. This allows the innovation likelihood to be computed and subsequently used for model classification via generalized Bayesian inference. Experimental results on real-world and simulated datasets demonstrate improved estimation accuracy and statistical consistency as well as robust classification performance across both low-data and large-data scenarios.

Robust state estimation is central to robotic autonomy, yet classical Kalman filters struggle with frequency-dependent disturbances and model mismatch such as sensor vibrations, electromagnetic interference, and periodic noise. Although Deep Kalman Filter (DKF) variants extend the Extended Kalman Filtering (EKF) framework by learning latent transitions, they lack explicit mechanisms to suppress band-limited noise components that typically corrupt sensor measurements in real-world scenarios. We introduce the Frequency-Weighted Neural Kalman Filter (FW-NKF), a unified hybrid approach that embeds a causal spectral-shaping operator into the Kalman measurement residual and jointly learns observation, and transition networks. By adapting both the filter spectrum and the latent state representation, FW-NKF attenuates the noise-dominated frequency bands while capturing complex residual structures. We conduct extensive experiments on four heterogeneous benchmarks, including chaotic systems such as multi-dimensional Lorenz systems and full-body inertial pose estimation, and find a reduction in localization error of up to 10% as well as marked improvements in orientation accuracy. Our ablation studies confirm that frequency weighting and deep latent-state modeling contribute to overall performance.

This paper introduces an estimation and control framework for dynamic landing of multi-rotor uncrewed aerial vehicles on moving platforms. The proposed method integrates nonlinear model predictive control with a real-time minimum-jerk trajectory planner that enforces a prescribed touchdown time, enabling consistent timing during the terminal descent. To enhance robustness in the presence of time-varying sensing quality, we utilize an adaptive unscented kalman filter that updates the process and measurement noise statistics online. In addition, we provide a reference feasibility analysis showing that minimum-jerk references induce bounded thrust and torque commands under standard tracking hypotheses. The proposed framework is evaluated in simulation and hardware experiments, and it is shown to achieve repeatable landings and improved platform velocity prediction accuracy relative to EKF/UKF-based methods.

State estimation is a fundamental problem in control and signal processing, for which the Kalman Filter provides an optimal solution under linear dynamics, Gaussian noise, and known noise covariances. However, these assumptions often fail in realistic sensing settings such as Doppler radar and LiDAR. In these cases, the optimal estimator is inherently nonlinear, which leads to systematic performance degradation. This creates a performance gap that cannot be eliminated by tuning the noise covariance parameters (i.e., the process and measurement noise in the Kalman Filter) alone. To address this limitation, we propose Kalman Evolve, a framework for discovering improved filtering algorithms by jointly optimizing both noise parameters and the update structure. Our approach leverages large language models (LLMs) as a structured prior over program space, enabling the generation of interpretable, non-affine modifications to the classical Kalman filter while preserving its recursive form. We provide analytical results establishing the suboptimality of affine estimators under common nonlinear sensing models, motivating the need for structure-aware updates. Across a range of synthetic and real-world tracking benchmarks, including Doppler radar, LiDAR-based localization, and pedestrian tracking, the discovered algorithms consistently improve over strong baselines such as the Optimized Kalman Filter, achieving up to 12% reduction in RMSE. These results suggest that optimizing the structure of the Kalman filter, rather than only its parameters, provides a practical and interpretable way to improve state estimation.

We present FusionCore, an open-source ROS 2 sensor fusion package that fuses IMU, wheel encoder odometry, GPS, and Visual SLAM pose into a single 100 Hz odometry stream using a 23-state Unscented Kalman Filter (UKF). The 23rd state is an online estimate of the wheel encoder's systematic yaw rate bias, identified through GPS heading cross-covariance and subtracted during GPS blackouts to reduce heading drift in coast mode. FusionCore also estimates gyroscope and accelerometer biases as explicit filter states, handles GPS natively in ECEF without a separate coordinate projection node, applies per-sensor Mahalanobis chi-squared outlier gating calibrated to measurement degrees of freedom, and adapts sensor noise covariance automatically from the innovation sequence. VSLAM pose fusion enables GPS-denied operation with any visual odometry or SLAM system, including automatic recovery from map reinitialization. We evaluate against robot_localization on twelve full-length sequences (55-92 min each) from the NCLT public dataset. FusionCore achieves lower Absolute Trajectory Error (ATE) on ten of twelve sequences, with improvements ranging from 1.2x to 22.2x on winning sequences. The robot_localization UKF diverges numerically on all twelve sequences. FusionCore is available at https://github.com/manankharwar/fusioncore under the Apache 2.0 license.

This paper presents Elevator-LIO, a LiDAR-inertial odometry framework designed to achieve continuous robot localization during elevator travel, thereby supporting cross-floor robotic tasks. To address the state-estimation problem in non-inertial frames, Elevator-LIO establishes a decoupled state-estimation model that separately models the robot motion relative to the elevator and the elevator motion itself, and embeds it into a mode-dependent iterated error-state Kalman filter framework. This framework degenerates to conventional LIO estimation in ordinary indoor environments, while enabling the propagation and constrained update of elevator-related states in elevator non-inertial environments, thereby achieving continuous and stable localization. An elevator mode manager detects elevator entry and exit events using LiDAR ranging statistics and estimated states, and introduces event-triggered zero-velocity and zero-acceleration updates when the elevator stops to suppress accumulated vertical drift. In addition, this paper adopts an adaptive voxel downsampling strategy to maintain a stable number of effective points under significant environmental scale changes. We conduct extensive experiments on 20 real-world sequences containing 79 elevator rides, including practical challenges such as large-scale spaces, long vertical travel, dynamic pedestrian interference, and mirror reflections. The results show that Elevator-LIO maintains continuous localization accuracy in all sequences, with terminal height error below 1 cm in 17 sequences. In contrast, existing representative localization systems perform poorly on these elevator sequences. Tests on the Hilti 2022/2023 datasets further show that the proposed method remains competitive in standard indoor scenarios. The project page is available at https://xiaofan4122.github.io/Elevator_LIO_Page/.

Unmanned Aerial Vehicles in dynamic environments face telemetry outages, structural vibrations, and regime-dependent noise that invalidate the stationary covariance assumptions of classical Kalman filters. The Sage-Husa Kalman Filter (SHKF) estimates noise statistics online, but its reliance on a static, scalar forgetting factor forces a strict compromise between steady-state stability and transient responsiveness. We introduce the N-Deep Recurrent Sage-Husa Filter (NDR-SHKF), which replaces this scalar parameter with a vector-valued memory attenuation policy learned by a hierarchical recurrent network operating on whitened innovation sequences. A bifurcated architecture routes shallow recurrent states to capture instantaneous sensor anomalies and deep states to encode sustained dynamic trends, while an auxiliary reconstruction objective prevents feature collapse. The complete filter, including recursive covariance updates, is trained end-to-end via backpropagation through time to directly minimize state estimation error. Evaluations on topologically distinct chaotic attractors demonstrate cross-domain generalization, outperforming purely data-driven baselines that diverge under out-of-distribution dynamics. Furthermore, evaluations on recorded real-world UAV flight datasets validate the framework's practical viability, demonstrating its capacity to bridge transitions into proprioceptive dead reckoning and outperform classical adaptive estimators during sensor outages.

Digital twins (DTs) rely on continuous synchronization between physical systems and their virtual counterparts through online parameter estimation under uncertainty. In many practical settings, however, this task is challenged by low observability, weak excitation, nonlinear dynamics, and noisy or biased measurements. In this work, we develop a new mathematical framework that integrates Weighted Flow Matching (WFM) generative modeling with physics-informed nonlinear filtering to enhance parameter estimation in DTs. WFM relies on dynamic reweighting of training samples, which guides the generative model toward parameter regimes most informative of the evolving system state. This generative component is tightly coupled with a physics-informed filtering architecture based on the Unscented Kalman Filter (UKF), yielding a unified DT framework that combines data-driven probability transport with physically consistent state and parameter estimation. The effectiveness of the new integrated framework is demonstrated within a spacecraft DT architecture, where stable moment of inertia estimation is achieved under uncertain and noisy sensing, with significant performance improvements over established approaches such as Extended Kalman Filtering (EKF) and Ensemble Kalman Filtering (EnKF). These results highlight the potential of weighted generative modeling as a core mechanism for real-time DT synchronization in operational and mission-critical systems.

A central obstacle in nonlinear Bayesian filtering is representing the belief distribution. Moment-based filters address this by propagating polynomial moments and reconstructing a density from them. Recent work completes the predict-update loop via the maximum-entropy (MaxEnt) principle, but each step requires the partition function and its gradient, both $n$-dimensional integrals whose cost scales exponentially, restricting the demonstrated MaxEnt moment filtering to $n \le 4$. We avoid the partition function entirely by combining score matching with Stein's identity. In our setting, score matching reduces the density fit to a single linear solve whose coefficients are assembled directly from the propagated moments. The same parameters then drive Stein's identity to close the moment hierarchy during prediction and to recover posterior moments after each Bayesian update, keeping the full predict-update loop free of partition function evaluation. The resulting Score Kalman Filter (SKF) reduces to the classical information-form Kalman filter as a special case and performs every step through linear algebra. On nonlinear coupled-oscillator networks, the SKF runs through $n=20$ and reports lower RMSE than the EKF, UKF, EnKF, and particle-filter baselines on the tested synthetic benchmarks.

This paper introduces the ensemble directional Kalman filter (EnDKF), an ensemble-based Kalman filtering approach for pose tracking that jointly estimates an object's position and attitude using ideas from directional statistics. The EnDKF integrates a unit-quaternion attitude representation to move beyond canonical Kalman filter mean and covariance assumptions that poorly capture directional uncertainty. Experiments on a synthetic constant-velocity constant-angular-velocity system and a digital-twin head-tracking scenario using the FoundationPose algorithm demonstrate a significant reduction in error as opposed to merely using measurements.

This paper presents a dual quaternion framework for 6-DOF visual target tracking that addresses key limitations of perspective-n-point (P$n$P) solvers: sensitivity to noise and outliers, and inability to propagate estimates through measurement dropouts. A nonlinear observability analysis is performed using a Lie algebraic approach, deriving sufficient conditions for local observability under two sensing modalities: relative position vector and unit vector measurements. For the unit vector case, the classical collinear feature point degeneracy of the perspective-three-point problem is recovered through rank analysis of the observability codistribution matrix, providing a control-theoretic interpretation of a previously geometric result. A dual quaternion Lie group unscented Kalman filter is then developed, directly modeling relative dynamics without assumptions about cooperative measurements or slowly-varying motion. Simulations demonstrate improved pose estimation accuracy and robustness to occlusions compared to an off-the-shelf P$n$P solver. Results are broadly applicable to visual-inertial navigation, simultaneous localization and mapping, and P$n$P solver development.

Accurate state estimation of nonlinear dynamical systems is fundamental to modern aerospace operations across air, sea, and space domains. Online tracking of adversarial unmanned aerial vehicles (UAVs) is especially challenging due to agile nonlinear motion, noisy and sparse sensor measurements, and unknown control inputs; conditions that violate key assumptions of classical Kalman filter variants and degrade estimation performance. Neural networks (NNs) can learn complex nonlinear relationships from data, but lack principled uncertainty quantification, which is critical for state estimation tasks where confidence bounds drive downstream decisions. We address this with Bayesian Neural Networks (BNNs), which model uncertainty through distributions over network weights and produce predictive means and uncertainties via Monte Carlo sampling. Building on this, we propose the Bayesian Neural Kalman Filter (BNKF): a hybrid framework coupling a trained BNN with a Kalman correction step for robust online UAV state estimation. Unlike related neural Kalman approaches, BNKF produces full state predictions and incorporates Bayesian uncertainty directly into covariance propagation, improving robustness under high noise conditions. We evaluate BNKF under varying radar noise levels and sampling rates using synthetic nonlinear UAV flight data. Five fold cross validation demonstrates that BNKF outperforms Extended and Unscented Kalman Filters in accuracy, precision, and truth containment under degraded sensing. An ensemble variant (BNKFe) further improves precision in high-noise edge cases at a slight accuracy tradeoff. Runtime analysis confirms minimal inference overhead, supporting real-time deployment feasibility.

State estimation is a fundamental requirement in robotics, where the accurate determination of a robot's state is essential for stable operation despite inherent process disturbances and sensor noise. Traditionally, this is achieved through Kalman filtering, providing a statistically optimal estimate by balancing predictive models with noisy measurements. In the context of robotic swarms, the challenge shifts from individual accuracy to collective coordination, where the integration of global dynamics can significantly enhance the precision of the entire group. Existing estimation techniques rely on centralized processing or heavy communication protocols to reach a global consensus, which are frequently impractical in real-world deployments. Here we show that a localized, "greedy" approach to distributed state estimation (termed "Greedy Kalman-Swarm") allows individual robots to leverage relative inter-robot sensing for improved accuracy without requiring full data availability or global communication. Simulations in communication-constrained environments show robots can effectively integrate all currently available neighbor data at each iteration to refine their internal states, yet remain robust and functional even when data is missing. This results in a performance profile that strikes a balance between the low overhead of independent estimation and the high accuracy of centralized systems, specifically under harsh or dynamic environmental conditions. Our results demonstrate that global state awareness can be emergent rather than enforced, providing a scalable framework for maintaining swarm cohesion in unpredictable terrains. We anticipate that this decentralized methodology will serve as a foundation for more resilient autonomous systems, particularly in search-and-rescue or space exploration missions where reliable, high-bandwidth communication cannot be guaranteed.

In Global Navigation Satellite System (GNSS)-denied underwater environments, individual unmanned underwater vehicles (UUVs) suffer from unbounded dead-reckoning drift, making collaborative navigation (CN) crucial for accurate state estimation. However, the severe communication delay inherent in underwater acoustic channels poses serious challenges to real-time state estimation. Traditional filters, such as Extended Kalman Filters (EKFs) or Unscented Kalman Filters (UKFs), usually block the main control loop while waiting for delayed data, or effectively discard Out-of-Sequence Measurements (OOSMs), resulting in serious drift. To address this, we propose an Asynchronous Two-Speed Kalman Filter (TSKF) enhanced by a novel projection mechanism, which we term Variational History Distillation (VHD). The proposed architecture decouples the estimation process into two parallel threads: a fast-rate thread that utilizes Gaussian Process (GP) compensated dead reckoning to guarantee high-frequency real-time control, and a slow-rate thread dedicated to processing asynchronously delayed collaborative information. By introducing a Finite-Length Circular State Buffer (FLCSB), the algorithm applies delayed measurements to their corresponding historical states, and utilizes a VHD-based projection to fast-forward the correction to the current time without computationally heavy recalculations. Simulation results demonstrate that the proposed TSKF maintains a trajectory error comparable to computationally intensive batch-optimization methods under severe delays (up to 30,s). Executing in sub-millisecond time, it significantly outperforms standard EKF/UKF. The results demonstrate an effective control, communication, and computing (3C) co-design that significantly enhances the resilience of autonomous marine automation systems.

Data assimilation (DA) estimates a dynamical system's state from noisy observations. Recent generative models like the ensemble score filter (EnSF) improve DA in high-dimensional nonlinear settings but are computationally expensive. We introduce the ensemble flow filter (EnFF), a training-free, flow matching (FM)-based framework that accelerates sampling and offers flexibility in flow design. EnFF uses Monte Carlo estimators for the marginal flow field, localized guidance for observation assimilation, and utilizes a novel flow path that exploits the Bayesian DA formulation. It generalizes classical filters such as the bootstrap particle filter and ensemble Kalman filter. Experiments on high-dimensional benchmarks demonstrate EnFF's improved cost-accuracy tradeoffs and scalability, highlighting FM's potential for efficient, scalable DA. Code is available at https://github.com/Utah-Math-Data-Science/Data-Assimilation-Flow-Matching.

Precise 3D state estimation in multi-object tracking (MOT) is critical for self-driving cars, particularly for objects occluded. Motion modeling in the Kalman filter with a constant motion assumption is widely used in MOT methods, but it neglects the continuous changes in objects' motion caused by traffic in urban environments. Although recent research introduces a multimodel Kalman filter that incorporates multiple motion models, these approaches incur significant computational overhead from the simultaneous processing of multiple models. To this end, this work introduces a motion-dynamics Kalman filter (MD-KF) that overcomes the constant-motion assumption while preserving the singularity of the motion model. MD-KF models the changes in objects' motion over successive measurements as Gaussian distributions, and adaptively adjusts a weighted motion model to account for these variations. MD-KF consistently outperforms constant and multimodel KF across multiple datasets with a significant reduction in computation latency compared to multimodel approaches. The proposed approach demonstrates its superiority in trajectory estimation during occlusion and state estimation stability for stationary objects.

Advanced Air Mobility (AAM) operations are planned to utilize strategic flight planning services that predict temporal uncertainties to validate flight plans against hazards such as weather cells, restricted airspaces, and CNS disruption areas. This paper presents a Kalman Filter-based uncertainty propagation method that models Flight Management System (FMS) correction behavior through a sigmoid-blended measurement noise covariance. The sigmoid formulation generalizes existing discrete FMS activation thresholds into a continuous, tunable function that smoothly transitions the filter's measurement noise based on progress toward each waypoint. When the measurement noise is high, due to an inverse relationship, the Kalman gain is small and thus uncertainty grows; as the aircraft nears a waypoint, measurement noise decreases as a function of progress, the Kalman gain increases, and state covariance contracts which models the FMS progressively correcting toward the planned trajectory. The approach is computationally efficient (up to two orders of magnitude faster than Monte Carlo methods), scales with control inputs, and is parametrically tunable for different classes of aircraft. The measurement noise covariance is calibrated using real Automatic Dependent Surveillance-Broadcast (ADS-B) data from commercial Instrument Flight Rules (IFR) flights serving as surrogates for future AAM operations, achieving coverage probability conservative relative to theoretical Gaussian predictions at the 1-sigma confidence level on a hold out verification dataset (N = 36). Parameter sensitivity analysis across multiple flight routes demonstrates robust behavior, and comparative evaluation against Monte Carlo and Linear Propagation methods contextualizes the method's computational and accuracy trade-offs.